Problem: Simplify; express your answer in exponential form. Assume $z\neq 0, a\neq 0$. $\dfrac{{(z^{-3})^{5}}}{{z^{-2}a^{-2}}}$
Solution: To start, try working on the numerator and the denominator independently. In the numerator, we have ${z^{-3}}$ to the exponent ${5}$ . Now ${-3 \times 5 = -15}$ , so ${(z^{-3})^{5} = z^{-15}}$ In the denominator, we can use the distributive property of exponents. ${z^{-2}a^{-2} = z^{-2}a^{-2}}$ Simplify using the same method from the numerator and put the entire equation together. $\dfrac{{(z^{-3})^{5}}}{{z^{-2}a^{-2}}} = \dfrac{{z^{-15}}}{{z^{-2}a^{-2}}}$ Break up the equation by variable and simplify. $\dfrac{{z^{-15}}}{{z^{-2}a^{-2}}} = \dfrac{{z^{-15}}}{{z^{-2}}} \cdot \dfrac{{1}}{{a^{-2}}} = z^{{-15} - {(-2)}} \cdot a^{- {(-2)}} = z^{-13}a^{2}$.